#133298

order	512
nilpotency class	4
isoclinism	isoclinic class of #133290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#2539	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

890 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(5,8)(9,10,13,18,14,19,11,16)(12,17,20,24,21,15,22,23);(1,2)(3,5)(4,6)(7,8)(10,15)(12,21)(16,23)(17,19)(18,24)(20,22);(1,3)(2,5)(4,7)(6,8)(9,11,14,13)(10,16,19,18)(12,20,21,22)(15,23,17,24)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40494);